I wish my wish would not be granted! - Douglas Hofstadter

"When a system of "meaningless" symbols has patterns in it that accurately track, or mirror, various phenomena in the world, then that tracking or mirroring imbues the symbols with some degree of meaning-indeed, such tracking or mirroring is no less and no more than what meaning is. Depending on how complex and subtle and reliable the tracking is, different degrees of meaningfulness arise."

There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any formal system.

Godel showed how a statement about any mathematical formal system (such as the assertion that Principia Mathematica is contradiction-free) can be translated into a mathematical statement inside number theory (the study of whole numbers). In other words, any metamathematical statement can be imported into mathematics, and in its new guise the statement simply asserts (as do all statements of number theory) that certain whole numbers have certain properties or relationships to each other. But on another level, it also has a vastly different meaning that, on its surface, seems as far removed from a statement of number theory as would be a sentence in a Dostoevsky novel.